void lm_lmdif( int m, int n, double* x, double* fvec, double ftol, double xtol,
double gtol, int maxfev, double epsfcn, double* diag, int mode,
double factor, int *info, int *nfev,
double* fjac, int* ipvt, double* qtf,
double* wa1, double* wa2, double* wa3, double* wa4,
lm_evaluate_ftype *evaluate, lm_print_ftype *printout,
void *data )
{
/*
* the purpose of lmdif is to minimize the sum of the squares of
* m nonlinear functions in n variables by a modification of
* the levenberg-marquardt algorithm. the user must provide a
* subroutine evaluate which calculates the functions. the jacobian
* is then calculated by a forward-difference approximation.
*
* the multi-parameter interface lm_lmdif is for users who want
* full control and flexibility. most users will be better off using
* the simpler interface lmfit provided above.
*
* the parameters are the same as in the legacy FORTRAN implementation,
* with the following exceptions:
* the old parameter ldfjac which gave leading dimension of fjac has
* been deleted because this C translation makes no use of two-
* dimensional arrays;
* the old parameter nprint has been deleted; printout is now controlled
* by the user-supplied routine *printout;
* the parameter field *data and the function parameters *evaluate and
* *printout have been added; they help avoiding global variables.
*
* parameters:
*
* m is a positive integer input variable set to the number
* of functions.
*
* n is a positive integer input variable set to the number
* of variables. n must not exceed m.
*
* x is an array of length n. on input x must contain
* an initial estimate of the solution vector. on output x
* contains the final estimate of the solution vector.
*
* fvec is an output array of length m which contains
* the functions evaluated at the output x.
*
* ftol is a nonnegative input variable. termination
* occurs when both the actual and predicted relative
* reductions in the sum of squares are at most ftol.
* therefore, ftol measures the relative error desired
* in the sum of squares.
*
* xtol is a nonnegative input variable. termination
* occurs when the relative error between two consecutive
* iterates is at most xtol. therefore, xtol measures the
* relative error desired in the approximate solution.
*
* gtol is a nonnegative input variable. termination
* occurs when the cosine of the angle between fvec and
* any column of the jacobian is at most gtol in absolute
* value. therefore, gtol measures the orthogonality
* desired between the function vector and the columns
* of the jacobian.
*
* maxfev is a positive integer input variable. termination
* occurs when the number of calls to lm_fcn is at least
* maxfev by the end of an iteration.
*
* epsfcn is an input variable used in determining a suitable
* step length for the forward-difference approximation. this
* approximation assumes that the relative errors in the
* functions are of the order of epsfcn. if epsfcn is less
* than the machine precision, it is assumed that the relative
* errors in the functions are of the order of the machine
* precision.
*
* diag is an array of length n. if mode = 1 (see below), diag is
* internally set. if mode = 2, diag must contain positive entries
* that serve as multiplicative scale factors for the variables.
*
* mode is an integer input variable. if mode = 1, the
* variables will be scaled internally. if mode = 2,
* the scaling is specified by the input diag. other
* values of mode are equivalent to mode = 1.
*
* factor is a positive input variable used in determining the
* initial step bound. this bound is set to the product of
* factor and the euclidean norm of diag*x if nonzero, or else
* to factor itself. in most cases factor should lie in the
* interval (.1,100.). 100. is a generally recommended value.
*
* info is an integer output variable that indicates the termination
* status of lm_lmdif as follows:
*
* info < 0 termination requested by user-supplied routine *evaluate;
*
* info = 0 improper input parameters;
*
* info = 1 both actual and predicted relative reductions
* in the sum of squares are at most ftol;
*
* info = 2 relative error between two consecutive iterates
* is at most xtol;
*
* info = 3 conditions for info = 1 and info = 2 both hold;
*
* info = 4 the cosine of the angle between fvec and any
* column of the jacobian is at most gtol in
* absolute value;
*
* info = 5 number of calls to lm_fcn has reached or
* exceeded maxfev;
*
* info = 6 ftol is too small. no further reduction in
* the sum of squares is possible;
*
* info = 7 xtol is too small. no further improvement in
* the approximate solution x is possible;
*
* info = 8 gtol is too small. fvec is orthogonal to the
* columns of the jacobian to machine precision;
*
* nfev is an output variable set to the number of calls to the
* user-supplied routine *evaluate.
*
* fjac is an output m by n array. the upper n by n submatrix
* of fjac contains an upper triangular matrix r with
* diagonal elements of nonincreasing magnitude such that
*
* t t t
* p *(jac *jac)*p = r *r,
*
* where p is a permutation matrix and jac is the final
* calculated jacobian. column j of p is column ipvt(j)
* (see below) of the identity matrix. the lower trapezoidal
* part of fjac contains information generated during
* the computation of r.
*
* ipvt is an integer output array of length n. ipvt
* defines a permutation matrix p such that jac*p = q*r,
* where jac is the final calculated jacobian, q is
* orthogonal (not stored), and r is upper triangular
* with diagonal elements of nonincreasing magnitude.
* column j of p is column ipvt(j) of the identity matrix.
*
* qtf is an output array of length n which contains
* the first n elements of the vector (q transpose)*fvec.
*
* wa1, wa2, and wa3 are work arrays of length n.
*
* wa4 is a work array of length m.
*
* the following parameters are newly introduced in this C translation:
*
* evaluate is the name of the subroutine which calculates the functions.
* a default implementation lm_evaluate_default is provided in lm_eval.c;
* alternatively, evaluate can be provided by a user calling program.
* it should be written as follows:
*
* void evaluate ( double* par, int m_dat, double* fvec,
* void *data, int *info )
* {
* // for ( i=0; i<m_dat; ++i )
* // calculate fvec[i] for given parameters par;
* // to stop the minimization,
* // set *info to a negative integer.
* }
*
* printout is the name of the subroutine which nforms about fit progress.
* a default implementation lm_print_default is provided in lm_eval.c;
* alternatively, printout can be provided by a user calling program.
* it should be written as follows:
*
* void printout ( int n_par, double* par, int m_dat, double* fvec,
* void *data, int iflag, int iter, int nfev )
* {
* // iflag : 0 (init) 1 (outer loop) 2(inner loop) -1(terminated)
* // iter : outer loop counter
* // nfev : number of calls to *evaluate
* }
*
* data is an input pointer to an arbitrary structure that is passed to
* evaluate. typically, it contains experimental data to be fitted.
*
*/
int i, iter, j;
double actred, delta, dirder, eps, fnorm, fnorm1, gnorm, par, pnorm,
prered, ratio, step, sum, temp, temp1, temp2, temp3, xnorm;
static double p1 = 0.1;
static double p5 = 0.5;
static double p25 = 0.25;
static double p75 = 0.75;
static double p0001 = 1.0e-4;
*nfev = 0; // function evaluation counter
iter = 1; // outer loop counter
par = 0; // levenberg-marquardt parameter
delta = 0; // just to prevent a warning (initialization within if-clause)
xnorm = 0; // dito
temp = MAX(epsfcn,LM_MACHEP);
eps = sqrt(temp); // used in calculating the Jacobian by forward differences
// *** check the input parameters for errors.
if ( (n <= 0) || (m < n) || (ftol < 0.)
|| (xtol < 0.) || (gtol < 0.) || (maxfev <= 0) || (factor <= 0.) )
{
*info = 0; // invalid parameter
return;
}
if ( mode == 2 ) /* scaling by diag[] */
{
for ( j=0; j<n; j++ ) /* check for nonpositive elements */
{
if ( diag[j] <= 0.0 )
{
*info = 0; // invalid parameter
return;
}
}
}
#if BUG
printf( "lmdif\n" );
#endif
// *** evaluate the function at the starting point and calculate its norm.
*info = 0;
(*evaluate)( x, m, fvec, data, info );
(*printout)( n, x, m, fvec, data, 0, 0, ++(*nfev) );
if ( *info < 0 ) return;
fnorm = lm_enorm(m,fvec);
// *** the outer loop.
do {
#if BUG
printf( "lmdif/ outer loop iter=%d nfev=%d fnorm=%.10e\n",
iter, *nfev, fnorm );
#endif
// O** calculate the jacobian matrix.
for ( j=0; j<n; j++ )
{
temp = x[j];
step = eps * fabs(temp);
if (step == 0.) step = eps;
x[j] = temp + step;
*info = 0;
(*evaluate)( x, m, wa4, data, info );
(*printout)( n, x, m, wa4, data, 1, iter, ++(*nfev) );
if ( *info < 0 ) return; // user requested break
x[j] = temp;
for ( i=0; i<m; i++ )
fjac[j*m+i] = (wa4[i] - fvec[i]) / step;
}
#if BUG>1
// DEBUG: print the entire matrix
for ( i=0; i<m; i++ )
{
for ( j=0; j<n; j++ )
printf( "%.5e ", y[j*m+i] );
printf( "\n" );
}
#endif
// O** compute the qr factorization of the jacobian.
lm_qrfac( m, n, fjac, 1, ipvt, wa1, wa2, wa3);
// O** on the first iteration ...
if (iter == 1)
{
if (mode != 2)
// ... scale according to the norms of the columns of the initial jacobian.
{
for ( j=0; j<n; j++ )
{
diag[j] = wa2[j];
if ( wa2[j] == 0. )
diag[j] = 1.;
}
}
// ... calculate the norm of the scaled x and
// initialize the step bound delta.
for ( j=0; j<n; j++ )
wa3[j] = diag[j] * x[j];
xnorm = lm_enorm( n, wa3 );
delta = factor*xnorm;
if (delta == 0.)
delta = factor;
}
// O** form (q transpose)*fvec and store the first n components in qtf.
for ( i=0; i<m; i++ )
wa4[i] = fvec[i];
for ( j=0; j<n; j++ )
{
temp3 = fjac[j*m+j];
if (temp3 != 0.)
{
sum = 0;
for ( i=j; i<m; i++ )
sum += fjac[j*m+i] * wa4[i];
temp = -sum / temp3;
for ( i=j; i<m; i++ )
wa4[i] += fjac[j*m+i] * temp;
}
fjac[j*m+j] = wa1[j];
qtf[j] = wa4[j];
}
// O** compute the norm of the scaled gradient and test for convergence.
gnorm = 0;
if ( fnorm != 0 )
{
for ( j=0; j<n; j++ )
{
if ( wa2[ ipvt[j] ] == 0 ) continue;
sum = 0.;
for ( i=0; i<=j; i++ )
sum += fjac[j*m+i] * qtf[i] / fnorm;
gnorm = MAX( gnorm, fabs(sum/wa2[ ipvt[j] ]) );
}
}
if ( gnorm <= gtol )
{
*info = 4;
return;
}
// O** rescale if necessary.
if ( mode != 2 )
{
for ( j=0; j<n; j++ )
diag[j] = MAX(diag[j],wa2[j]);
}
// O** the inner loop.
do {
#if BUG
printf( "lmdif/ inner loop iter=%d nfev=%d\n", iter, *nfev );
#endif
// OI* determine the levenberg-marquardt parameter.
lm_lmpar( n,fjac,m,ipvt,diag,qtf,delta,&par,wa1,wa2,wa3,wa4 );
// OI* store the direction p and x + p. calculate the norm of p.
for ( j=0; j<n; j++ )
{
wa1[j] = -wa1[j];
wa2[j] = x[j] + wa1[j];
wa3[j] = diag[j]*wa1[j];
}
pnorm = lm_enorm(n,wa3);
// OI* on the first iteration, adjust the initial step bound.
if ( *nfev<= 1+n ) // bug corrected by J. Wuttke in 2004
delta = MIN(delta,pnorm);
// OI* evaluate the function at x + p and calculate its norm.
*info = 0;
(*evaluate)( wa2, m, wa4, data, info );
(*printout)( n, x, m, wa4, data, 2, iter, ++(*nfev) );
if ( *info < 0 ) return; // user requested break
fnorm1 = lm_enorm(m,wa4);
#if BUG
printf( "lmdif/ pnorm %.10e fnorm1 %.10e fnorm %.10e"
" delta=%.10e par=%.10e\n",
pnorm, fnorm1, fnorm, delta, par );
#endif
// OI* compute the scaled actual reduction.
if ( p1*fnorm1 < fnorm )
actred = 1 - SQR( fnorm1/fnorm );
else
actred = -1;
// OI* compute the scaled predicted reduction and
// the scaled directional derivative.
for ( j=0; j<n; j++ )
{
wa3[j] = 0;
for ( i=0; i<=j; i++ )
wa3[i] += fjac[j*m+i]*wa1[ ipvt[j] ];
}
temp1 = lm_enorm(n,wa3) / fnorm;
temp2 = sqrt(par) * pnorm / fnorm;
prered = SQR(temp1) + 2 * SQR(temp2);
dirder = - ( SQR(temp1) + SQR(temp2) );
// OI* compute the ratio of the actual to the predicted reduction.
ratio = prered!=0 ? actred/prered : 0;
#if BUG
printf( "lmdif/ actred=%.10e prered=%.10e ratio=%.10e"
" sq(1)=%.10e sq(2)=%.10e dd=%.10e\n",
actred, prered, prered!=0 ? ratio : 0.,
SQR(temp1), SQR(temp2), dirder );
#endif
// OI* update the step bound.
if (ratio <= p25)
{
if (actred >= 0.)
temp = p5;
else
temp = p5*dirder/(dirder + p5*actred);
if ( p1*fnorm1 >= fnorm || temp < p1 )
temp = p1;
delta = temp * MIN(delta,pnorm/p1);
par /= temp;
}
else if ( par == 0. || ratio >= p75 )
{
delta = pnorm/p5;
par *= p5;
}
// OI* test for successful iteration...
if (ratio >= p0001)
{
// ... successful iteration. update x, fvec, and their norms.
for ( j=0; j<n; j++ )
{
x[j] = wa2[j];
wa2[j] = diag[j]*x[j];
}
for ( i=0; i<m; i++ )
fvec[i] = wa4[i];
xnorm = lm_enorm(n,wa2);
fnorm = fnorm1;
iter++;
}
#if BUG
else {
printf( "ATTN: iteration considered unsuccessful\n" );
}
#endif
// OI* tests for convergence ( otherwise *info = 1, 2, or 3 )
*info = 0; // do not terminate (unless overwritten by nonzero value)
if ( fabs(actred) <= ftol && prered <= ftol && p5*ratio <= 1 )
*info = 1;
if (delta <= xtol*xnorm)
*info += 2;
if ( *info != 0)
return;
// OI* tests for termination and stringent tolerances.
if ( *nfev >= maxfev)
*info = 5;
if ( fabs(actred) <= LM_MACHEP &&
prered <= LM_MACHEP && p5*ratio <= 1 )
*info = 6;
if (delta <= LM_MACHEP*xnorm)
*info = 7;
if (gnorm <= LM_MACHEP)
*info = 8;
if ( *info != 0)
return;
// OI* end of the inner loop. repeat if iteration unsuccessful.
} while (ratio < p0001);
// O** end of the outer loop.
} while (1);
}
void lm_lmdif( int m, int n, double *x, double *fvec, double ftol,
double xtol, double gtol, int maxfev, double epsfcn,
double *diag, int mode, double factor, int *info, int *nfev,
double *fjac, int *ipvt, double *qtf, double *wa1,
double *wa2, double *wa3, double *wa4,
void (*evaluate) (const double *par, int m_dat, const void *data,
double *fvec, int *info),
void (*printout) (int n_par, const double *par, int m_dat,
const void *data, const double *fvec,
int printflags, int iflag, int iter, int nfev),
int printflags, const void *data )
{
/*
* The purpose of lmdif is to minimize the sum of the squares of
* m nonlinear functions in n variables by a modification of
* the levenberg-marquardt algorithm. The user must provide a
* subroutine evaluate which calculates the functions. The jacobian
* is then calculated by a forward-difference approximation.
*
* The multi-parameter interface lm_lmdif is for users who want
* full control and flexibility. Most users will be better off using
* the simpler interface lmmin provided above.
*
* Parameters:
*
* m is a positive integer input variable set to the number
* of functions.
*
* n is a positive integer input variable set to the number
* of variables; n must not exceed m.
*
* x is an array of length n. On input x must contain an initial
* estimate of the solution vector. On OUTPUT x contains the
* final estimate of the solution vector.
*
* fvec is an OUTPUT array of length m which contains
* the functions evaluated at the output x.
*
* ftol is a nonnegative input variable. Termination occurs when
* both the actual and predicted relative reductions in the sum
* of squares are at most ftol. Therefore, ftol measures the
* relative error desired in the sum of squares.
*
* xtol is a nonnegative input variable. Termination occurs when
* the relative error between two consecutive iterates is at
* most xtol. Therefore, xtol measures the relative error desired
* in the approximate solution.
*
* gtol is a nonnegative input variable. Termination occurs when
* the cosine of the angle between fvec and any column of the
* jacobian is at most gtol in absolute value. Therefore, gtol
* measures the orthogonality desired between the function vector
* and the columns of the jacobian.
*
* maxfev is a positive integer input variable. Termination
* occurs when the number of calls to lm_fcn is at least
* maxfev by the end of an iteration.
*
* epsfcn is an input variable used in choosing a step length for
* the forward-difference approximation. The relative errors in
* the functions are assumed to be of the order of epsfcn.
*
* diag is an array of length n. If mode = 1 (see below), diag is
* internally set. If mode = 2, diag must contain positive entries
* that serve as multiplicative scale factors for the variables.
*
* mode is an integer input variable. If mode = 1, the
* variables will be scaled internally. If mode = 2,
* the scaling is specified by the input diag.
*
* factor is a positive input variable used in determining the
* initial step bound. This bound is set to the product of
* factor and the euclidean norm of diag*x if nonzero, or else
* to factor itself. In most cases factor should lie in the
* interval (0.1,100.0). Generally, the value 100.0 is recommended.
*
* info is an integer OUTPUT variable that indicates the termination
* status of lm_lmdif as follows:
*
* info < 0 termination requested by user-supplied routine *evaluate;
*
* info = 0 fnorm almost vanishing;
*
* info = 1 both actual and predicted relative reductions
* in the sum of squares are at most ftol;
*
* info = 2 relative error between two consecutive iterates
* is at most xtol;
*
* info = 3 conditions for info = 1 and info = 2 both hold;
*
* info = 4 the cosine of the angle between fvec and any
* column of the jacobian is at most gtol in
* absolute value;
*
* info = 5 number of calls to lm_fcn has reached or
* exceeded maxfev;
*
* info = 6 ftol is too small: no further reduction in
* the sum of squares is possible;
*
* info = 7 xtol is too small: no further improvement in
* the approximate solution x is possible;
*
* info = 8 gtol is too small: fvec is orthogonal to the
* columns of the jacobian to machine precision;
*
* info =10 improper input parameters;
*
* nfev is an OUTPUT variable set to the number of calls to the
* user-supplied routine *evaluate.
*
* fjac is an OUTPUT m by n array. The upper n by n submatrix
* of fjac contains an upper triangular matrix r with
* diagonal elements of nonincreasing magnitude such that
*
* pT*(jacT*jac)*p = rT*r
*
* (NOTE: T stands for matrix transposition),
*
* where p is a permutation matrix and jac is the final
* calculated jacobian. Column j of p is column ipvt(j)
* (see below) of the identity matrix. The lower trapezoidal
* part of fjac contains information generated during
* the computation of r.
*
* ipvt is an integer OUTPUT array of length n. It defines a
* permutation matrix p such that jac*p = q*r, where jac is
* the final calculated jacobian, q is orthogonal (not stored),
* and r is upper triangular with diagonal elements of
* nonincreasing magnitude. Column j of p is column ipvt(j)
* of the identity matrix.
*
* qtf is an OUTPUT array of length n which contains
* the first n elements of the vector (q transpose)*fvec.
*
* wa1, wa2, and wa3 are work arrays of length n.
*
* wa4 is a work array of length m, used among others to hold
* residuals from evaluate.
*
* evaluate points to the subroutine which calculates the
* m nonlinear functions. Implementations should be written as follows:
*
* void evaluate( double* par, int m_dat, void *data,
* double* fvec, int *info )
* {
* // for ( i=0; i<m_dat; ++i )
* // calculate fvec[i] for given parameters par;
* // to stop the minimization,
* // set *info to a negative integer.
* }
*
* printout points to the subroutine which informs about fit progress.
* Call with printout=0 if no printout is desired.
* Call with printout=lm_printout_std to use the default implementation.
*
* printflags is passed to printout.
*
* data is an input pointer to an arbitrary structure that is passed to
* evaluate. Typically, it contains experimental data to be fitted.
*
*/
int i, iter, j;
double actred, delta, dirder, eps, fnorm, fnorm1, gnorm, par, pnorm,
prered, ratio, step, sum, temp, temp1, temp2, temp3, xnorm;
static double p1 = 0.1;
static double p0001 = 1.0e-4;
*nfev = 0; /* function evaluation counter */
iter = 0; /* outer loop counter */
par = 0; /* levenberg-marquardt parameter */
delta = 0; /* to prevent a warning (initialization within if-clause) */
xnorm = 0; /* ditto */
temp = MAX(epsfcn, LM_MACHEP);
eps = sqrt(temp); /* for calculating the Jacobian by forward differences */
/*** lmdif: check input parameters for errors. ***/
if ((n <= 0) || (m < n) || (ftol < 0.)
|| (xtol < 0.) || (gtol < 0.) || (maxfev <= 0) || (factor <= 0.)) {
*info = 10; // invalid parameter
return;
}
if (mode == 2) { /* scaling by diag[] */
for (j = 0; j < n; j++) { /* check for nonpositive elements */
if (diag[j] <= 0.0) {
*info = 10; // invalid parameter
return;
}
}
}
#ifdef LMFIT_DEBUG_MESSAGES
printf("lmdif\n");
#endif
/*** lmdif: evaluate function at starting point and calculate norm. ***/
*info = 0;
(*evaluate) (x, m, data, fvec, info);
++(*nfev);
if( printout )
(*printout) (n, x, m, data, fvec, printflags, 0, 0, *nfev);
if (*info < 0)
return;
fnorm = lm_enorm(m, fvec);
if( fnorm <= LM_DWARF ){
*info = 0;
return;
}
/*** lmdif: the outer loop. ***/
do {
#ifdef LMFIT_DEBUG_MESSAGES
printf("lmdif/ outer loop iter=%d nfev=%d fnorm=%.10e\n",
iter, *nfev, fnorm);
#endif
/*** outer: calculate the Jacobian. ***/
for (j = 0; j < n; j++) {
temp = x[j];
step = MAX(eps*eps, eps * fabs(temp));
x[j] = temp + step; /* replace temporarily */
*info = 0;
(*evaluate) (x, m, data, wa4, info);
++(*nfev);
if( printout )
(*printout) (n, x, m, data, wa4, printflags, 1, iter, *nfev);
if (*info < 0)
return; /* user requested break */
for (i = 0; i < m; i++)
fjac[j*m+i] = (wa4[i] - fvec[i]) / step;
x[j] = temp; /* restore */
}
#ifdef LMFIT_DEBUG_MATRIX
/* print the entire matrix */
for (i = 0; i < m; i++) {
for (j = 0; j < n; j++)
printf("%.5e ", fjac[j*m+i]);
printf("\n");
}
#endif
/*** outer: compute the qr factorization of the Jacobian. ***/
lm_qrfac(m, n, fjac, 1, ipvt, wa1, wa2, wa3);
/* return values are ipvt, wa1=rdiag, wa2=acnorm */
if (!iter) {
/* first iteration only */
if (mode != 2) {
/* diag := norms of the columns of the initial Jacobian */
for (j = 0; j < n; j++) {
diag[j] = wa2[j];
if (wa2[j] == 0.)
diag[j] = 1.;
}
}
/* use diag to scale x, then calculate the norm */
for (j = 0; j < n; j++)
wa3[j] = diag[j] * x[j];
xnorm = lm_enorm(n, wa3);
/* initialize the step bound delta. */
delta = factor * xnorm;
if (delta == 0.)
delta = factor;
} else {
if (mode != 2) {
for (j = 0; j < n; j++)
diag[j] = MAX( diag[j], wa2[j] );
}
}
/*** outer: form (q transpose)*fvec and store first n components in qtf. ***/
for (i = 0; i < m; i++)
wa4[i] = fvec[i];
for (j = 0; j < n; j++) {
temp3 = fjac[j*m+j];
if (temp3 != 0.) {
sum = 0;
for (i = j; i < m; i++)
sum += fjac[j*m+i] * wa4[i];
temp = -sum / temp3;
for (i = j; i < m; i++)
wa4[i] += fjac[j*m+i] * temp;
}
fjac[j*m+j] = wa1[j];
qtf[j] = wa4[j];
}
/*** outer: compute norm of scaled gradient and test for convergence. ***/
gnorm = 0;
for (j = 0; j < n; j++) {
if (wa2[ipvt[j]] == 0)
continue;
sum = 0.;
for (i = 0; i <= j; i++)
sum += fjac[j*m+i] * qtf[i];
gnorm = MAX( gnorm, fabs( sum / wa2[ipvt[j]] / fnorm ) );
}
if (gnorm <= gtol) {
*info = 4;
return;
}
/*** the inner loop. ***/
do {
#ifdef LMFIT_DEBUG_MESSAGES
printf("lmdif/ inner loop iter=%d nfev=%d\n", iter, *nfev);
#endif
/*** inner: determine the levenberg-marquardt parameter. ***/
lm_lmpar( n, fjac, m, ipvt, diag, qtf, delta, &par,
wa1, wa2, wa4, wa3 );
/* used return values are fjac (partly), par, wa1=x, wa3=diag*x */
for (j = 0; j < n; j++)
wa2[j] = x[j] - wa1[j]; /* new parameter vector ? */
pnorm = lm_enorm(n, wa3);
/* at first call, adjust the initial step bound. */
if (*nfev <= 1+n)
delta = MIN(delta, pnorm);
/*** inner: evaluate the function at x + p and calculate its norm. ***/
*info = 0;
(*evaluate) (wa2, m, data, wa4, info);
++(*nfev);
if( printout )
(*printout) (n, wa2, m, data, wa4, printflags, 2, iter, *nfev);
if (*info < 0)
return; /* user requested break. */
fnorm1 = lm_enorm(m, wa4);
#ifdef LMFIT_DEBUG_MESSAGES
printf("lmdif/ pnorm %.10e fnorm1 %.10e fnorm %.10e"
" delta=%.10e par=%.10e\n",
pnorm, fnorm1, fnorm, delta, par);
#endif
/*** inner: compute the scaled actual reduction. ***/
if (p1 * fnorm1 < fnorm)
actred = 1 - SQR(fnorm1 / fnorm);
else
actred = -1;
/*** inner: compute the scaled predicted reduction and
the scaled directional derivative. ***/
for (j = 0; j < n; j++) {
wa3[j] = 0;
for (i = 0; i <= j; i++)
wa3[i] -= fjac[j*m+i] * wa1[ipvt[j]];
}
temp1 = lm_enorm(n, wa3) / fnorm;
temp2 = sqrt(par) * pnorm / fnorm;
prered = SQR(temp1) + 2 * SQR(temp2);
dirder = -(SQR(temp1) + SQR(temp2));
/*** inner: compute the ratio of the actual to the predicted reduction. ***/
ratio = prered != 0 ? actred / prered : 0;
#ifdef LMFIT_DEBUG_MESSAGES
printf("lmdif/ actred=%.10e prered=%.10e ratio=%.10e"
" sq(1)=%.10e sq(2)=%.10e dd=%.10e\n",
actred, prered, prered != 0 ? ratio : 0.,
SQR(temp1), SQR(temp2), dirder);
#endif
/*** inner: update the step bound. ***/
if (ratio <= 0.25) {
if (actred >= 0.)
temp = 0.5;
else
temp = 0.5 * dirder / (dirder + 0.5 * actred);
if (p1 * fnorm1 >= fnorm || temp < p1)
temp = p1;
delta = temp * MIN(delta, pnorm / p1);
par /= temp;
} else if (par == 0. || ratio >= 0.75) {
delta = pnorm / 0.5;
par *= 0.5;
}
/*** inner: test for successful iteration. ***/
if (ratio >= p0001) {
/* yes, success: update x, fvec, and their norms. */
for (j = 0; j < n; j++) {
x[j] = wa2[j];
wa2[j] = diag[j] * x[j];
}
for (i = 0; i < m; i++)
fvec[i] = wa4[i];
xnorm = lm_enorm(n, wa2);
fnorm = fnorm1;
iter++;
}
#ifdef LMFIT_DEBUG_MESSAGES
else {
printf("ATTN: iteration considered unsuccessful\n");
}
#endif
/*** inner: test for convergence. ***/
if( fnorm<=LM_DWARF ){
*info = 0;
return;
}
*info = 0;
if (fabs(actred) <= ftol && prered <= ftol && 0.5 * ratio <= 1)
*info = 1;
if (delta <= xtol * xnorm)
*info += 2;
if (*info != 0)
return;
/*** inner: tests for termination and stringent tolerances. ***/
if (*nfev >= maxfev){
*info = 5;
return;
}
if (fabs(actred) <= LM_MACHEP &&
prered <= LM_MACHEP && 0.5 * ratio <= 1){
*info = 6;
return;
}
if (delta <= LM_MACHEP * xnorm){
*info = 7;
return;
}
if (gnorm <= LM_MACHEP){
*info = 8;
return;
}
/*** inner: end of the loop. repeat if iteration unsuccessful. ***/
} while (ratio < p0001);
/*** outer: end of the loop. ***/
} while (1);
} /*** lm_lmdif. ***/
void lmmin(const int n, double* x, const int m, const void* data,
void (*evaluate)(const double* par, const int m_dat,
const void* data, double* fvec, int* userbreak),
const lm_control_struct* C, lm_status_struct* S)
{
int j, i;
double actred, dirder, fnorm, fnorm1, gnorm, pnorm, prered, ratio, step,
sum, temp, temp1, temp2, temp3;
/*** Initialize internal variables. ***/
int maxfev = C->patience * (n+1);
int inner_success; /* flag for loop control */
double lmpar = 0; /* Levenberg-Marquardt parameter */
double delta = 0;
double xnorm = 0;
double eps = sqrt(MAX(C->epsilon, LM_MACHEP)); /* for forward differences */
int nout = C->n_maxpri == -1 ? n : MIN(C->n_maxpri, n);
/* Reinterpret C->msgfile=NULL as stdout (which is unavailable for
compile-time initialization of lm_control_double and similar). */
FILE* msgfile = C->msgfile ? C->msgfile : stdout;
/*** Default status info; must be set before first return statement. ***/
S->outcome = 0; /* status code */
S->userbreak = 0;
S->nfev = 0; /* function evaluation counter */
/*** Check input parameters for errors. ***/
if (n <= 0) {
fprintf(stderr, "lmmin: invalid number of parameters %i\n", n);
S->outcome = 10;
return;
}
if (m < n) {
fprintf(stderr, "lmmin: number of data points (%i) "
"smaller than number of parameters (%i)\n",
m, n);
S->outcome = 10;
return;
}
if (C->ftol < 0 || C->xtol < 0 || C->gtol < 0) {
fprintf(stderr,
"lmmin: negative tolerance (at least one of %g %g %g)\n",
C->ftol, C->xtol, C->gtol);
S->outcome = 10;
return;
}
if (maxfev <= 0) {
fprintf(stderr, "lmmin: nonpositive function evaluations limit %i\n",
maxfev);
S->outcome = 10;
return;
}
if (C->stepbound <= 0) {
fprintf(stderr, "lmmin: nonpositive stepbound %g\n", C->stepbound);
S->outcome = 10;
return;
}
if (C->scale_diag != 0 && C->scale_diag != 1) {
fprintf(stderr, "lmmin: logical variable scale_diag=%i, "
"should be 0 or 1\n",
C->scale_diag);
S->outcome = 10;
return;
}
/*** Allocate work space. ***/
/* Allocate total workspace with just one system call */
char* ws;
if ((ws = (char *)malloc((2*m + 5*n + m*n) * sizeof(double) +
n * sizeof(int))) == NULL) {
S->outcome = 9;
return;
}
/* Assign workspace segments. */
char* pws = ws;
double* fvec = (double*)pws;
pws += m * sizeof(double) / sizeof(char);
double* diag = (double*)pws;
pws += n * sizeof(double) / sizeof(char);
double* qtf = (double*)pws;
pws += n * sizeof(double) / sizeof(char);
double* fjac = (double*)pws;
pws += n * m * sizeof(double) / sizeof(char);
double* wa1 = (double*)pws;
pws += n * sizeof(double) / sizeof(char);
double* wa2 = (double*)pws;
pws += n * sizeof(double) / sizeof(char);
double* wa3 = (double*)pws;
pws += n * sizeof(double) / sizeof(char);
double* wf = (double*)pws;
pws += m * sizeof(double) / sizeof(char);
int* Pivot = (int*)pws;
//pws += n * sizeof(int) / sizeof(char);
/* Initialize diag. */
if (!C->scale_diag)
for (j = 0; j < n; j++)
diag[j] = 1;
/*** Evaluate function at starting point and calculate norm. ***/
if (C->verbosity) {
fprintf(msgfile, "lmmin start ");
lm_print_pars(nout, x, msgfile);
}
(*evaluate)(x, m, data, fvec, &(S->userbreak));
if (C->verbosity > 4)
for (i = 0; i < m; ++i)
fprintf(msgfile, " fvec[%4i] = %18.8g\n", i, fvec[i]);
S->nfev = 1;
if (S->userbreak)
goto terminate;
fnorm = lm_enorm(m, fvec);
if (C->verbosity)
fprintf(msgfile, " fnorm = %18.8g\n", fnorm);
if (!isfinite(fnorm)) {
S->outcome = 12; /* nan */
goto terminate;
} else if (fnorm <= LM_DWARF) {
S->outcome = 0; /* sum of squares almost zero, nothing to do */
goto terminate;
}
/*** The outer loop: compute gradient, then descend. ***/
for (int outer = 0;; ++outer) {
/** Calculate the Jacobian. **/
for (j = 0; j < n; j++) {
temp = x[j];
step = MAX(eps * eps, eps * fabs(temp));
x[j] += step; /* replace temporarily */
(*evaluate)(x, m, data, wf, &(S->userbreak));
++(S->nfev);
if (S->userbreak)
goto terminate;
for (i = 0; i < m; i++)
fjac[j*m+i] = (wf[i] - fvec[i]) / step;
x[j] = temp; /* restore */
}
if (C->verbosity >= 10) {
/* print the entire matrix */
printf("\nlmmin Jacobian\n");
for (i = 0; i < m; i++) {
printf(" ");
for (j = 0; j < n; j++)
printf("%.5e ", fjac[j*m+i]);
printf("\n");
}
}
/** Compute the QR factorization of the Jacobian. **/
/* fjac is an m by n array. The upper n by n submatrix of fjac is made
* to contain an upper triangular matrix R with diagonal elements of
* nonincreasing magnitude such that
*
* P^T*(J^T*J)*P = R^T*R
*
* (NOTE: ^T stands for matrix transposition),
*
* where P is a permutation matrix and J is the final calculated
* Jacobian. Column j of P is column Pivot(j) of the identity matrix.
* The lower trapezoidal part of fjac contains information generated
* during the computation of R.
*
* Pivot is an integer array of length n. It defines a permutation
* matrix P such that jac*P = Q*R, where jac is the final calculated
* Jacobian, Q is orthogonal (not stored), and R is upper triangular
* with diagonal elements of nonincreasing magnitude. Column j of P
* is column Pivot(j) of the identity matrix.
*/
lm_qrfac(m, n, fjac, Pivot, wa1, wa2, wa3);
/* return values are Pivot, wa1=rdiag, wa2=acnorm */
/** Form Q^T * fvec, and store first n components in qtf. **/
for (i = 0; i < m; i++)
wf[i] = fvec[i];
for (j = 0; j < n; j++) {
temp3 = fjac[j*m+j];
if (temp3 != 0) {
sum = 0;
for (i = j; i < m; i++)
sum += fjac[j*m+i] * wf[i];
temp = -sum / temp3;
for (i = j; i < m; i++)
wf[i] += fjac[j*m+i] * temp;
}
fjac[j*m+j] = wa1[j];
qtf[j] = wf[j];
}
/** Compute norm of scaled gradient and detect degeneracy. **/
gnorm = 0;
for (j = 0; j < n; j++) {
if (wa2[Pivot[j]] == 0)
continue;
sum = 0;
for (i = 0; i <= j; i++)
sum += fjac[j*m+i] * qtf[i];
gnorm = MAX(gnorm, fabs(sum / wa2[Pivot[j]] / fnorm));
}
if (gnorm <= C->gtol) {
S->outcome = 4;
goto terminate;
}
/** Initialize or update diag and delta. **/
if (!outer) { /* first iteration only */
if (C->scale_diag) {
/* diag := norms of the columns of the initial Jacobian */
for (j = 0; j < n; j++)
diag[j] = wa2[j] ? wa2[j] : 1;
/* xnorm := || D x || */
for (j = 0; j < n; j++)
wa3[j] = diag[j] * x[j];
xnorm = lm_enorm(n, wa3);
if (C->verbosity >= 2) {
fprintf(msgfile, "lmmin diag ");
lm_print_pars(nout, x, msgfile); // xnorm
fprintf(msgfile, " xnorm = %18.8g\n", xnorm);
}
/* Only now print the header for the loop table. */
if (C->verbosity >= 3) {
fprintf(msgfile, " o i lmpar prered"
" ratio dirder delta"
" pnorm fnorm");
for (i = 0; i < nout; ++i)
fprintf(msgfile, " p%i", i);
fprintf(msgfile, "\n");
}
} else {
xnorm = lm_enorm(n, x);
}
if (!isfinite(xnorm)) {
S->outcome = 12; /* nan */
goto terminate;
}
/* Initialize the step bound delta. */
if (xnorm)
delta = C->stepbound * xnorm;
else
delta = C->stepbound;
} else {
if (C->scale_diag) {
for (j = 0; j < n; j++)
diag[j] = MAX(diag[j], wa2[j]);
}
}
/** The inner loop. **/
int inner = 0;
do {
/** Determine the Levenberg-Marquardt parameter. **/
lm_lmpar(n, fjac, m, Pivot, diag, qtf, delta, &lmpar,
wa1, wa2, wf, wa3);
/* used return values are fjac (partly), lmpar, wa1=x, wa3=diag*x */
/* Predict scaled reduction. */
pnorm = lm_enorm(n, wa3);
if (!isfinite(pnorm)) {
S->outcome = 12; /* nan */
goto terminate;
}
temp2 = lmpar * SQR(pnorm / fnorm);
for (j = 0; j < n; j++) {
wa3[j] = 0;
for (i = 0; i <= j; i++)
wa3[i] -= fjac[j*m+i] * wa1[Pivot[j]];
}
temp1 = SQR(lm_enorm(n, wa3) / fnorm);
if (!isfinite(temp1)) {
S->outcome = 12; /* nan */
goto terminate;
}
prered = temp1 + 2*temp2;
dirder = -temp1 + temp2; /* scaled directional derivative */
/* At first call, adjust the initial step bound. */
if (!outer && pnorm < delta)
delta = pnorm;
/** Evaluate the function at x + p. **/
for (j = 0; j < n; j++)
wa2[j] = x[j] - wa1[j];
(*evaluate)(wa2, m, data, wf, &(S->userbreak));
++(S->nfev);
if (S->userbreak)
goto terminate;
fnorm1 = lm_enorm(m, wf);
if (!isfinite(fnorm1)) {
S->outcome = 12; /* nan */
goto terminate;
}
/** Evaluate the scaled reduction. **/
/* Actual scaled reduction. */
actred = 1 - SQR(fnorm1 / fnorm);
/* Ratio of actual to predicted reduction. */
ratio = prered ? actred / prered : 0;
if (C->verbosity == 2) {
fprintf(msgfile, "lmmin (%i:%i) ", outer, inner);
lm_print_pars(nout, wa2, msgfile); // fnorm1,
} else if (C->verbosity >= 3) {
printf("%3i %2i %9.2g %9.2g %14.6g"
" %9.2g %10.3e %10.3e %21.15e",
outer, inner, lmpar, prered, ratio,
dirder, delta, pnorm, fnorm1);
for (i = 0; i < nout; ++i)
fprintf(msgfile, " %16.9g", wa2[i]);
fprintf(msgfile, "\n");
}
/* Update the step bound. */
if (ratio <= 0.25) {
if (actred >= 0)
temp = 0.5;
else if (actred > -99) /* -99 = 1-1/0.1^2 */
temp = MAX(dirder / (2*dirder + actred), 0.1);
else
temp = 0.1;
delta = temp * MIN(delta, pnorm / 0.1);
lmpar /= temp;
} else if (ratio >= 0.75) {
delta = 2 * pnorm;
lmpar *= 0.5;
} else if (!lmpar) {
delta = 2 * pnorm;
}
/** On success, update solution, and test for convergence. **/
inner_success = ratio >= 1e-4;
if (inner_success) {
/* Update x, fvec, and their norms. */
if (C->scale_diag) {
for (j = 0; j < n; j++) {
x[j] = wa2[j];
wa2[j] = diag[j] * x[j];
}
} else {
for (j = 0; j < n; j++)
x[j] = wa2[j];
}
for (i = 0; i < m; i++)
fvec[i] = wf[i];
xnorm = lm_enorm(n, wa2);
if (!isfinite(xnorm)) {
S->outcome = 12; /* nan */
goto terminate;
}
fnorm = fnorm1;
}
/* Convergence tests. */
S->outcome = 0;
if (fnorm <= LM_DWARF)
goto terminate; /* success: sum of squares almost zero */
/* Test two criteria (both may be fulfilled). */
if (fabs(actred) <= C->ftol && prered <= C->ftol && ratio <= 2)
S->outcome = 1; /* success: x almost stable */
if (delta <= C->xtol * xnorm)
S->outcome += 2; /* success: sum of squares almost stable */
if (S->outcome != 0) {
goto terminate;
}
/** Tests for termination and stringent tolerances. **/
if (S->nfev >= maxfev) {
S->outcome = 5;
goto terminate;
}
if (fabs(actred) <= LM_MACHEP && prered <= LM_MACHEP &&
ratio <= 2) {
S->outcome = 6;
goto terminate;
}
if (delta <= LM_MACHEP * xnorm) {
S->outcome = 7;
goto terminate;
}
if (gnorm <= LM_MACHEP) {
S->outcome = 8;
goto terminate;
}
/** End of the inner loop. Repeat if iteration unsuccessful. **/
++inner;
} while (!inner_success);
}; /*** End of the outer loop. ***/
terminate:
S->fnorm = lm_enorm(m, fvec);
if (C->verbosity >= 2)
printf("lmmin outcome (%i) xnorm %g ftol %g xtol %g\n", S->outcome,
xnorm, C->ftol, C->xtol);
if (C->verbosity & 1) {
fprintf(msgfile, "lmmin final ");
lm_print_pars(nout, x, msgfile); // S->fnorm,
fprintf(msgfile, " fnorm = %18.8g\n", S->fnorm);
}
if (S->userbreak) /* user-requested break */
S->outcome = 11;
/*** Deallocate the workspace. ***/
free(ws);
} /*** lmmin. ***/
void lmmin(
const int n, double *const x, const int m, const double* y,
const void *const data,
void (*const evaluate)(
const double *const par, const int m_dat, const void *const data,
double *const fvec, int *const userbreak),
const lm_control_struct *const C, lm_status_struct *const S)
{
int j, i;
double actred, dirder, fnorm, fnorm1, gnorm, pnorm,
prered, ratio, step, sum, temp, temp1, temp2, temp3;
static double p1 = 0.1, p0001 = 1.0e-4;
int maxfev = C->patience * (n+1);
int inner_success; /* flag for loop control */
double lmpar = 0; /* Levenberg-Marquardt parameter */
double delta = 0;
double xnorm = 0;
double eps = sqrt(MAX(C->epsilon, LM_MACHEP)); /* for forward differences */
int nout = C->n_maxpri==-1 ? n : MIN(C->n_maxpri, n);
/* The workaround msgfile=NULL is needed for default initialization */
FILE* msgfile = C->msgfile ? C->msgfile : stdout;
/* Default status info; must be set ahead of first return statements */
S->outcome = 0; /* status code */
S->userbreak = 0;
S->nfev = 0; /* function evaluation counter */
/*** Check input parameters for errors. ***/
if ( n < 0 ) {
fprintf(stderr, "lmmin: invalid number of parameters %i\n", n);
S->outcome = 10; /* invalid parameter */
return;
}
if (m < n) {
fprintf(stderr, "lmmin: number of data points (%i) "
"smaller than number of parameters (%i)\n", m, n);
S->outcome = 10;
return;
}
if (C->ftol < 0 || C->xtol < 0 || C->gtol < 0) {
fprintf(stderr,
"lmmin: negative tolerance (at least one of %g %g %g)\n",
C->ftol, C->xtol, C->gtol);
S->outcome = 10;
return;
}
if (maxfev <= 0) {
fprintf(stderr, "lmmin: nonpositive function evaluations limit %i\n",
maxfev);
S->outcome = 10;
return;
}
if (C->stepbound <= 0) {
fprintf(stderr, "lmmin: nonpositive stepbound %g\n", C->stepbound);
S->outcome = 10;
return;
}
if (C->scale_diag != 0 && C->scale_diag != 1) {
fprintf(stderr, "lmmin: logical variable scale_diag=%i, "
"should be 0 or 1\n", C->scale_diag);
S->outcome = 10;
return;
}
/*** Allocate work space. ***/
/* Allocate total workspace with just one system call */
char *ws;
if ( ( ws = static_cast<char *>(malloc(
(2*m+5*n+m*n)*sizeof(double) + n*sizeof(int)) ) ) == NULL ) {
S->outcome = 9;
return;
}
/* Assign workspace segments. */
char *pws = ws;
double *fvec = (double*) pws; pws += m * sizeof(double)/sizeof(char);
double *diag = (double*) pws; pws += n * sizeof(double)/sizeof(char);
double *qtf = (double*) pws; pws += n * sizeof(double)/sizeof(char);
double *fjac = (double*) pws; pws += n*m*sizeof(double)/sizeof(char);
double *wa1 = (double*) pws; pws += n * sizeof(double)/sizeof(char);
double *wa2 = (double*) pws; pws += n * sizeof(double)/sizeof(char);
double *wa3 = (double*) pws; pws += n * sizeof(double)/sizeof(char);
double *wf = (double*) pws; pws += m * sizeof(double)/sizeof(char);
int *ipvt = (int*) pws; /*pws += n * sizeof(int) /sizeof(char);*/
/* Initialize diag */ // TODO: check whether this is still needed
if (!C->scale_diag) {
for (j = 0; j < n; j++)
diag[j] = 1.;
}
/*** Evaluate function at starting point and calculate norm. ***/
if( C->verbosity&1 )
fprintf(msgfile, "lmmin start (ftol=%g gtol=%g xtol=%g)\n",
C->ftol, C->gtol, C->xtol);
if( C->verbosity&2 )
lm_print_pars(nout, x, msgfile);
(*evaluate)(x, m, data, fvec, &(S->userbreak));
if( C->verbosity&8 )
{
if (y) {
for( i=0; i<m; ++i )
fprintf(msgfile, " i, f, y-f: %4i %18.8g %18.8g\n",
i, fvec[i], y[i]-fvec[i]);
} else {
for( i=0; i<m; ++i )
fprintf(msgfile, " i, f: %4i %18.8g\n", i, fvec[i]);
}
}
S->nfev = 1;
if ( S->userbreak )
goto terminate;
if ( n == 0 ) {
S->outcome = 13; /* won't fit */
goto terminate;
}
fnorm = lm_fnorm(m, fvec, y);
if( C->verbosity&2 )
fprintf(msgfile, " fnorm = %24.16g\n", fnorm);
if( !isfinite(fnorm) ){
if( C->verbosity )
fprintf(msgfile, "nan case 1\n");
S->outcome = 12; /* nan */
goto terminate;
} else if( fnorm <= LM_DWARF ){
S->outcome = 0; /* sum of squares almost zero, nothing to do */
goto terminate;
}
/*** The outer loop: compute gradient, then descend. ***/
for( int outer=0; ; ++outer ) {
/*** [outer] Calculate the Jacobian. ***/
for (j = 0; j < n; j++) {
temp = x[j];
step = MAX(eps*eps, eps * fabs(temp));
x[j] += step; /* replace temporarily */
(*evaluate)(x, m, data, wf, &(S->userbreak));
++(S->nfev);
if ( S->userbreak )
goto terminate;
for (i = 0; i < m; i++)
fjac[j*m+i] = (wf[i] - fvec[i]) / step;
x[j] = temp; /* restore */
}
if ( C->verbosity&16 ) {
/* print the entire matrix */
printf("Jacobian\n");
for (i = 0; i < m; i++) {
printf(" ");
for (j = 0; j < n; j++)
printf("%.5e ", fjac[j*m+i]);
printf("\n");
}
}
/*** [outer] Compute the QR factorization of the Jacobian. ***/
/* fjac is an m by n array. The upper n by n submatrix of fjac
* is made to contain an upper triangular matrix R with diagonal
* elements of nonincreasing magnitude such that
*
* P^T*(J^T*J)*P = R^T*R
*
* (NOTE: ^T stands for matrix transposition),
*
* where P is a permutation matrix and J is the final calculated
* Jacobian. Column j of P is column ipvt(j) of the identity matrix.
* The lower trapezoidal part of fjac contains information generated
* during the computation of R.
*
* ipvt is an integer array of length n. It defines a permutation
* matrix P such that jac*P = Q*R, where jac is the final calculated
* Jacobian, Q is orthogonal (not stored), and R is upper triangular
* with diagonal elements of nonincreasing magnitude. Column j of P
* is column ipvt(j) of the identity matrix.
*/
lm_qrfac(m, n, fjac, ipvt, wa1, wa2, wa3);
/* return values are ipvt, wa1=rdiag, wa2=acnorm */
/*** [outer] Form Q^T * fvec, and store first n components in qtf. ***/
if (y)
for (i = 0; i < m; i++)
wf[i] = fvec[i] - y[i];
else
for (i = 0; i < m; i++)
wf[i] = fvec[i];
for (j = 0; j < n; j++) {
temp3 = fjac[j*m+j];
if (temp3 != 0) {
sum = 0;
for (i = j; i < m; i++)
sum += fjac[j*m+i] * wf[i];
temp = -sum / temp3;
for (i = j; i < m; i++)
wf[i] += fjac[j*m+i] * temp;
}
fjac[j*m+j] = wa1[j];
qtf[j] = wf[j];
}
/*** [outer] Compute norm of scaled gradient and detect degeneracy. ***/
gnorm = 0;
for (j = 0; j < n; j++) {
if (wa2[ipvt[j]] == 0)
continue;
sum = 0;
for (i = 0; i <= j; i++)
sum += fjac[j*m+i] * qtf[i];
gnorm = MAX(gnorm, fabs( sum / wa2[ipvt[j]] / fnorm ));
}
if (gnorm <= C->gtol) {
S->outcome = 4;
goto terminate;
}
/*** [outer] Initialize / update diag and delta. ***/
if ( !outer ) {
/* first iteration only */
if (C->scale_diag) {
/* diag := norms of the columns of the initial Jacobian */
for (j = 0; j < n; j++)
diag[j] = wa2[j] ? wa2[j] : 1;
/* xnorm := || D x || */
for (j = 0; j < n; j++)
wa3[j] = diag[j] * x[j];
xnorm = lm_enorm(n, wa3);
} else {
xnorm = lm_enorm(n, x);
}
if( !isfinite(xnorm) ){
if( C->verbosity )
fprintf(msgfile, "nan case 2\n");
S->outcome = 12; /* nan */
goto terminate;
}
/* initialize the step bound delta. */
if ( xnorm )
delta = C->stepbound * xnorm;
else
delta = C->stepbound;
/* only now print the header for the loop table */
if( C->verbosity&2 ) {
fprintf(msgfile, " #o #i lmpar prered actred"
" ratio dirder delta"
" pnorm fnorm");
for (i = 0; i < nout; ++i)
fprintf(msgfile, " p%i", i);
fprintf(msgfile, "\n");
}
} else {
if (C->scale_diag) {
for (j = 0; j < n; j++)
diag[j] = MAX( diag[j], wa2[j] );
}
}
/*** The inner loop. ***/
int inner = 0;
do {
/*** [inner] Determine the Levenberg-Marquardt parameter. ***/
lm_lmpar(n, fjac, m, ipvt, diag, qtf, delta, &lmpar,
wa1, wa2, wf, wa3);
/* used return values are fjac (partly), lmpar, wa1=x, wa3=diag*x */
/* predict scaled reduction */
pnorm = lm_enorm(n, wa3);
if( !isfinite(pnorm) ){
if( C->verbosity )
fprintf(msgfile, "nan case 3\n");
S->outcome = 12; /* nan */
goto terminate;
}
temp2 = lmpar * SQR( pnorm / fnorm );
for (j = 0; j < n; j++) {
wa3[j] = 0;
for (i = 0; i <= j; i++)
wa3[i] -= fjac[j*m+i] * wa1[ipvt[j]];
}
temp1 = SQR( lm_enorm(n, wa3) / fnorm );
if( !isfinite(temp1) ){
if( C->verbosity )
fprintf(msgfile, "nan case 4\n");
S->outcome = 12; /* nan */
goto terminate;
}
prered = temp1 + 2 * temp2;
dirder = -temp1 + temp2; /* scaled directional derivative */
/* at first call, adjust the initial step bound. */
if ( !outer && !inner && pnorm < delta )
delta = pnorm;
/*** [inner] Evaluate the function at x + p. ***/
for (j = 0; j < n; j++)
wa2[j] = x[j] - wa1[j];
(*evaluate)( wa2, m, data, wf, &(S->userbreak) );
++(S->nfev);
if ( S->userbreak )
goto terminate;
fnorm1 = lm_fnorm(m, wf, y);
// exceptionally, for this norm we do not test for infinity
// because we can deal with it without terminating.
/*** [inner] Evaluate the scaled reduction. ***/
/* actual scaled reduction (supports even the case fnorm1=infty) */
if (p1 * fnorm1 < fnorm)
actred = 1 - SQR(fnorm1 / fnorm);
else
actred = -1;
/* ratio of actual to predicted reduction */
ratio = prered ? actred/prered : 0;
if( C->verbosity&32 ) {
if (y) {
for( i=0; i<m; ++i )
fprintf(msgfile, " i, f, y-f: %4i %18.8g %18.8g\n",
i, fvec[i], y[i]-fvec[i]);
} else {
for( i=0; i<m; ++i )
fprintf(msgfile, " i, f, y-f: %4i %18.8g\n",
i, fvec[i]);
}
}
if( C->verbosity&2 ) {
printf("%3i %2i %9.2g %9.2g %9.2g %14.6g"
" %9.2g %10.3e %10.3e %21.15e",
outer, inner, lmpar, prered, actred, ratio,
dirder, delta, pnorm, fnorm1);
for (i = 0; i < nout; ++i)
fprintf(msgfile, " %16.9g", wa2[i]);
fprintf(msgfile, "\n");
}
/* update the step bound */
if (ratio <= 0.25) {
if (actred >= 0)
temp = 0.5;
else
temp = 0.5 * dirder / (dirder + 0.5 * actred);
if (p1 * fnorm1 >= fnorm || temp < p1)
temp = p1;
delta = temp * MIN(delta, pnorm / p1);
lmpar /= temp;
} else if (lmpar == 0 || ratio >= 0.75) {
delta = 2 * pnorm;
lmpar *= 0.5;
}
/*** [inner] On success, update solution, and test for convergence. ***/
inner_success = ratio >= p0001;
if ( inner_success ) {
/* update x, fvec, and their norms */
if (C->scale_diag) {
for (j = 0; j < n; j++) {
x[j] = wa2[j];
wa2[j] = diag[j] * x[j];
}
} else {
for (j = 0; j < n; j++)
x[j] = wa2[j];
}
for (i = 0; i < m; i++)
fvec[i] = wf[i];
xnorm = lm_enorm(n, wa2);
if( !isfinite(xnorm) ){
if( C->verbosity )
fprintf(msgfile, "nan case 6\n");
S->outcome = 12; /* nan */
goto terminate;
}
fnorm = fnorm1;
}
/* convergence tests */
S->outcome = 0;
if( fnorm<=LM_DWARF )
goto terminate; /* success: sum of squares almost zero */
/* test two criteria (both may be fulfilled) */
if (fabs(actred) <= C->ftol && prered <= C->ftol && ratio <= 2)
S->outcome = 1; /* success: x almost stable */
if (delta <= C->xtol * xnorm)
S->outcome += 2; /* success: sum of squares almost stable */
if (S->outcome != 0) {
goto terminate;
}
/*** [inner] Tests for termination and stringent tolerances. ***/
if ( S->nfev >= maxfev ){
S->outcome = 5;
goto terminate;
}
if ( fabs(actred) <= LM_MACHEP &&
prered <= LM_MACHEP && ratio <= 2 ){
S->outcome = 6;
goto terminate;
}
if ( delta <= LM_MACHEP*xnorm ){
S->outcome = 7;
goto terminate;
}
if ( gnorm <= LM_MACHEP ){
S->outcome = 8;
goto terminate;
}
/*** [inner] End of the loop. Repeat if iteration unsuccessful. ***/
++inner;
} while ( !inner_success );
/*** [outer] End of the loop. ***/
};
terminate:
S->fnorm = lm_fnorm(m, fvec, y);
if( C->verbosity&1 )
fprintf(msgfile, "lmmin terminates with outcome %i\n", S->outcome);
if( C->verbosity&2 )
lm_print_pars(nout, x, msgfile);
if( C->verbosity&8 ) {
if (y) {
for( i=0; i<m; ++i )
fprintf(msgfile, " i, f, y-f: %4i %18.8g %18.8g\n",
i, fvec[i], y[i]-fvec[i] );
} else {
for( i=0; i<m; ++i )
fprintf(msgfile, " i, f, y-f: %4i %18.8g\n", i, fvec[i]);
}
}
if( C->verbosity&2 )
fprintf(msgfile, " fnorm=%24.16g xnorm=%24.16g\n", S->fnorm, xnorm);
if ( S->userbreak ) /* user-requested break */
S->outcome = 11;
/*** Deallocate the workspace. ***/
free(ws);
} /*** lmmin. ***/
